Maria Colonna Laboratori Nazionali del Sud (Catania)

Testing the behavior of n-rich systems away from normal density

Eurorib’ 10June 6-11, 2010 --- Lamoura

Equation of State (EoS) of asymmetric nuclear matter the nuclear energy density functionals, effective interactions

ˆ H

E

EeffH

Self-consistent MF calculations (and extensions) are a powerful framework to understand the structure of medium-heavy nuclei.

Source: F.Gulminelli

In this context relativistic <=> non-relativistic …only a matter of functional

Isoscalar, spin, isospin densities, currents …

Widely employed in the astrophysical context (modelization of neutron stars and supernova explosion)

The largest uncertainties concern the isovector part of the nuclear interaction : The symmetry energy

E/A (ρ) = Es(ρ) + Esym(ρ) β² β=(N-Z)/A

asy-stiff

asy-soft

zoom at low density

asy-soft

asy-stiff

C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book)

Esym(ρ) = J γ = L/(3J)

Esym = E/A (β=1) – E/A(β=0)

Often used parametrization:

asy-soft, asy-stiff

)/( 0potsymE

Focus on Esym at low density

The crust-core transition density decreases with L

Nuclear structure

Nuclear astrophysics

Correlation between n-skin and L

Properties of n-rich nuclei depend onlow-density Esym (because of surface effects !)

Nuclei- neutron starconnection !

M.Centelles et al, PRL(2009)I.Vidana et al., PRC80(2009)

Isospin effects in reaction mechanisms at Fermi energiesIsospin effects in reaction mechanisms at Fermi energies

Symmetry energy parameterizations are implemented into transport codes (Stochastic Mean Field - SMF) and confronted to experimental data for specific reaction mechanisms and related observables Chomaz,Colonna, Randrup Phys. Rep. 389 (2004)

Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005)

asy-soft

asy-stiff

Parametrizations used in SMF simulations

Esympot =

18 r (2 – r) SKM*(soft)18 r stiff18 (2r2 )/(1+r) stiff (superstiff)

r = ρ/ρ0

Transient states of nuclear matterin several conditions !

γ~0.6

γ~1

Reactions between systems with different N/ZIsospin diffusion (in the low density interface) is driven by the symmetry energy Information on Esym at low density

INDRA data: Ni + Ni, Ni + Au @ 52, 74 MeV/A

ISOSPIN TRANSPORT AT FERMI ENERGIESISOSPIN TRANSPORT AT FERMI ENERGIES

1(PLF)

2(TLF)

Reaction plane1) If x = N/Z or f(N/Z) Isospin equilibration2) Contact time measured by kinetic energy dissipation Symmetry energy

x1,2(t) – xm = (x1,2 – x m) e-t/τ xm = (x1 + x2)/2

t contact timeτ dissipation time for observable x

Path towards equilibrium of the observable x

Galichet et al.,Phys. Rev. C79, 064615 (2009)

How to access the N/Z of the PLF ? Isotopic content of light charged particle emission as a function of the dissipated energy

Exchange of energy, mass,isospin between 1 and 2

Calculations:- N/Z increases with the centrality ofcollision for the two systems and energies(For Ni + Ni pre-equilibrium effects)- In Ni + Au systems more isospin diffusion for asy-soft (as expected)- (N/Z)CP linearly correlated to (N/Z)QP

PLF

PLF

CP

CP

PLF

PLF

CP

CP

N/ZCP

-- stiff (γ=1) + SIMON-- soft(γ=0.6) + SIMON

SMF transport calculations:N/Z of the PLF (Quasi-Projectile)Squares: soft Stars: stiff

After statistical decay :

N = Σi Ni , Z = Σi Zi

Charged particles: Z=1-4

forward n-n c.m.

forward PLF

Comparison with data

Data: open points higher than full points(n-rich mid-rapidity particles) Isospin equilibration reached for Ediss/Ecm = 0.7-0.8 ? (open and full dots converge) Data fall between the two calculations

R1,2(t) = (x1,2(t) – xm) / |x1,2 – xm| R1,2 = ±e-t/τ

Xm

X2

X1

x1,2(t) – xm = (x1,2 – x m) e-t/τ xm = (x1 + x2)/2

Path towards equilibrium of the observable x B. Tsang et al. PRL 92 (2004)

1(PLF)

2(TLF)

Isospin transport ratio R Isospin transport ratio R

N/Z of largest fragment

yred

Ni + Ni@ 15,40AMeV

P.N

apol

itan

i et

al.,

PR

C(2

010)

τ Esym

Focus on Esym below normal density

Strength of PDR

Mass formula

Neutron skin thickness

Li, Lombardo, Schulze, Zuo, PRC77, 034316 (2008)

Microscopic BHF calculations

Galichet et al.,(2009)

A.C

arb

one

et a

l., P

RC

(R)

(20

10)

an

d r

ef.s

th

erei

n

Nuclear reactions:Isospin diffusion

Tsang et al., PRL(2009)

GMR (Li et al, PRL 2007)

Pre-equilibrium dipole emission

Need to enlarge the systematics of data (and calculations) to validate the current interpretation and the extraction of Esym (consensus on Esym ~(ρ/ρ0)γ with γ~0.6-1 at low density)

Still large uncertainties at high density (FAIR, NICA, RIKEN, …)

V.Baran (NIPNE HH,Bucharest) M.Di Toro, C.Rizzo, J.Rizzo, (LNS, Catania)M.Zielinska-Pfabe (Smith College) H.H.Wolter (Munich)E.Galichet, P.Napolitani (IPN, Orsay)

Conclusions

( , , ) ( , , ) ( , , ) ( ) ( )K r p t K r p t p p r r r t t

fWWfdt

df

Ensemble average

Langevin: randomwalk in phase-space

Transport model: Semi-classical approach to the many-body problemTransport model: Semi-classical approach to the many-body problem

Time evolution of the one-body distribution function ( , , )f r p t

Boltzmann

),,()(),(),(),( tprKfKprffhprft

LangevinVlasov

Vlasov Boltzmann Langevin

)(2

)(2

fUm

pfh

i

i

Vlasov: mean field

Boltzmann: average collision term

( ) ( ) NNf i f i

dp p E E

d

3 3 32 1 2

2 1' 2'3 3 3( , ) (12 1 2 )

d p d p d pW r p f f f w

h h h Loss term

D(p,p’,r)

SMF model : fluctuations projected onto ordinary space density fluctuations δρ

Fluctuation variance: σ2f = <δfδf>

D(p,p’,r) w

Probes of the symmetry energy (at low density)

Isospin diffusion J.Rizzo et al, NPA (2008)

Pre-equilibrium dipole oscillationV.Baran et al, PRC79, 021603 (2009).

Isospin distillation (liquid-gas)

asy-stiff - - -asy-soft

M.Colonna et alPRC78,064618(2008)

Optical potentials (isospin & momentum dependence of forces)

Li & Lombardo, PRC78,047603(2008)

• Trippa, Colò, Vigezzi PRC77(2008)061304

• P.Danielewicz J.Lee nucl-th/08073743

• A.Klimkiewicz et al PRC76(2007)051603

• B.Tsang et al PRL102(2009)122701

Pygmy dipole

mass formula

Isospin

diffusion

GDR

L=3

dE

sym/d

P0 =

L/3

BHF

Fragment N/Z,Central collisions

GDR

Constraints on Esym

• Li,Lombardo et al PRC77(2008)034316

• Galichet,Colonna et al PRC79(2009)064615

• M.Colonna et alPRC78,064618(2008)

Symmetry energy at ρ0 (normal density)

C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book)

E/A (ρ) = Es(ρ) + Esym(ρ) β² β=(N-Z)/A

data

Momentum dependence

effective mass different for protons and neutrons

1

2

*

1

k

U

k

m

m

m qq

m*n < m*p

m*n > m*p

Asy-softAsy-stiff

n

p

* 10%np

m

2.0

Often used parametrization:

asy-soft, asy-stiff

)/( 0potsymE

Symmetry energy and mass splittingSymmetry energy and mass splitting

asy-stiff

asy-soft

zoom at low density

asy-soft

asy-stiff

2pn

Lane

UUU

Lane potential Symmetry potential

Ediss

1(PLF)

2(TLF)

The charge of the reconstructed PLF is in reasonnable agreement with the data

The dissipated energy is well correlated to the impact parameter

Sorting variable and PLF propertiesSorting variable and PLF properties

Galichet et al.,Phys. Rev. C79, 064615 (2009)